G. Isac Departement de Mathematiques, College Militaire Royal, St. Jean, Q&bee, Canada JOJ IR
A.B.Németh Institutul de Matematica, Str. Republicii Nr. 37, 3400 Cluj-Napoca, Roumanie
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G. Isac, A.B. Nemeth, Every generating isotone projection cone is latticial and correct, J. Math. Anal. Appl., 147 (1990) no. 1, pp. 53-62
doi: 10.1016/0022-247X(90)90383-Q
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4. G. ISAC, A. B. NEMETH, Monotonicity of metric projections onto positive cones of ordered euclidean spaces, Arch. Math. 46 (1986) 5688576; Corrigendum, Arch. Math. 49 (1987), 367-368.
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6. G. ISAC AND A. B. NEMETH, Ordered Hilbert spaces, preprint, 1987.
7. C. W. MTARTHUR, In what spaces is every closed normal cone regular? Proc. Edinburgh Math. Soc. (2) 17 (1970), 121-125.
8. J. MOREAU, Decomposition orthogonale d’un espace hilbertien selon deux cones mutuellement pollaires, C. R. Acad. Sci. Paris Ser Math. 225 (1962), 238-240.
9. F. RIESZ, Sur quelques notions fondamentales dans la theorie g&r&ale des operations lineaires, Mat. Term~szet~ l&es. 56 (1937), 145; Ann. qf Math. 41 (1940), 174-206.
10. E. H. ZARANTONELLO, Projections on convex sets in Hilbert space and spectral theory, in “Contributions to Nonlinear Functional Analysis” (E. H. Zarantonello, Ed.), pp. 237424, Academic Press, New York, 1971.
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Paper (preprint) in HTML form
1990-Nemeth-Every generating isotone projection cone is latticial
Every Generating Isotone Projection Cone Is Latticial and Correct
G. IsacDépartement de Mathématiques, Collège Militaire Royal, St. Jean, Québec, Canada J0J lR0
ANDA. B. NémethInstitutul de Matematica, Str. Republicii Nr. 37, 3400 Cluj-Napoca, RoumanieSubmitted by E. Stanley Lee
Received September 12, 1988
1. Introduction
In the last decade, ordered topological vector spaces have been studied a great deal, but ordered Hilbert spaces have mysteriously been neglected.
It is interesting to remark that the ordered Hilbert space has interesting and deep properties and important applications in areas such as Quantum Mechanics [3], Potential Theory, Analytic Manifold Theory, the study of positivity of Green's functions, Nonlinear Analysis, the study of the Complementarity Problem [6], etc.
In connection with the study of the Complementarity Problem we introduced in [5,4] the concept of isotone projection cone. Certainly, the isotone projection cones have applications in Numerical Analysis, Variational Inequalities, and Optimization by projection methods.
Let (H,(::)(H,\langle\rangle,)beaHilbertspaceandletK sub H) be a Hilbert space and let K \subset H be a pointed convex cone; that is, (1) K+K sub KK+K \subset K, (2) (AA lambda inR_(+))(lambda K sub K)\left(\forall \lambda \in R_{+}\right)(\lambda K \subset K), and (3) K nn(-K)={0}K \cap(-K)=\{0\}.
We denote by " <=\leqslant " the order defined on HH by KK, that is, x <= y<=>y-x in Kx \leqslant y \Leftrightarrow y-x \in K.
If KK is closed it is well known [10] that for every x in Hx \in H there exists a unique element P_(K)(x)in KP_{K}(x) \in K (called the projection of xx onto KK ) such that
||x-P_(K)(x)|| <= ||x-y||,quad" for every "y in K.\left\|x-P_{K}(x)\right\| \leqslant\|x-y\|, \quad \text { for every } y \in K .
The following problem has never previously been considered in the mathematical literature.
When is P_(K)P_{K} monotone increasing (isotone) with respect to the order defined by KK ?
So, we say that KK is an isotone projection if for every x,y in H,x <= yx, y \in H, x \leqslant y implies that P_(K)(x) <= P_(K)(y)P_{K}(x) \leqslant P_{K}(y) [5,4].
We are interested in characterizing by a geometric or analytic property isotone projection cones.
In Euclidean space we know a characterization of isotone projection cones by a necessary and sufficient condition [4,5] but in infinite dimensional Hilbert spaces this problem is not solved.
In this note we prove that if K sub HK \subset H is a generating isotone projection cone in HH, then it is latticial and correct.
But, if K sub HK \subset H is a closed convex cone, generating, latticial, and correct, is it an isotone projection? We do not know.
2. Definitions and Main Results
Let (H,(::)(H,\langle\rangle,)beaHilbertspace.IfK sub H) be a Hilbert space. If K \subset H is a closed convex cone, then for every x in Hx \in H the projection P_(K)(x)P_{K}(x) can be characterized by the relation
{:(1)(:x-P_(K)(x),P_(K)(x)-y:) >= 0;quad AA y in K(" see [10, Lemma 1.1] ").:}\begin{equation*}
\left\langle x-P_{K}(x), P_{K}(x)-y\right\rangle \geqslant 0 ; \quad \forall y \in K(\text { see [10, Lemma 1.1] }) . \tag{1}
\end{equation*}
The dual of KK is K^(**)={y in H∣(:x,y:) >= 0,AA x in K}K^{*}=\{y \in H \mid\langle x, y\rangle \geqslant 0, \forall x \in K\}. K^(**)K^{*} is a closed convex cone and if KK is generating, that is, if K-K=HK-K=H, then K^(**)K^{*} is a pointed convex cone.
Besides (1) the theorem of Moreau will be the other basic instrument in our proof.
The polar of KK is K^(@)=-K^(**)K^{\circ}=-K^{*}. If KK is closed then K=(K^(@))^(@)K=\left(K^{\circ}\right)^{\circ}.
If KK and QQ are two closed convex cones in HH then we say that KK and QQ are mutually polar if K=Q^(@)K=Q^{\circ} (which implies K^(@)=QK^{\circ}=Q ).
Theorem (Moreau [8]). If KK and QQ are two mutually polar convex cones in HH and x,y,z in Hx, y, z \in H then the following statements are equivalent:
(i) z=x+y,x in K,y in Qz=x+y, x \in K, y \in Q, and (:x,y:)=0\langle x, y\rangle=0,
(ii) x=P_(K)(z)x=P_{K}(z) and y=P_(Q)(z)y=P_{Q}(z).
Every convex cone will be supposed pointed.
Definition 1. A convex cone K sub HK \subset H is called an isotone projection if
{:(2)K" is closed, ":}\begin{equation*}
K \text { is closed, } \tag{2}
\end{equation*}
whenever v-u in Kv-u \in K it follows that P_(K)(v)-P_(K)(u)in KP_{K}(v)-P_{K}(u) \in K.
Remark. If we consider on HH the order " <=\leqslant " defined by KK then condition (3) of Definition 1 says that P_(K)P_{K} is isotone with respect to the order defined by KK, whence the reason for our terminology.
Observe that (3) in Definition 1 is a purely geometric definition and it does not depend on the position of KK in HH, while an order theoretic definition requires KK to be a "cone with vertex at the origin," i.e., a cone in the sense of our above definition.
The space HH ordered by KK as above will be called a vector lattice and KK will be called latticial if every pair of elements uu and vv in HH has a least upper bound denoted by u vv vu \vee v.
Let K sub HK \subset H be a convex cone. A convex cone F sub KF \subset K is called a face of KK if x in K,y in Fx \in K, y \in F, and y-x in Ky-x \in K imply x in Fx \in F.
(Note that this notion is less restrictive than the notion of face used in [10], which is in fact exposed face.)
Definition 2. The cone KK in HH will be called correct if for each of face FF, we have
where bar(sp)F\overline{s p} F is the closed linear span of FF.
Remark. The above notion of correct cone is related to that of perfect cone which turns out to coincide with correct cone for a self-dual cone (see Proposition 1 in [1]).
In [4] we gave a characterization of isotone projection cones in Euclidean spaces.
In [5] we considered problems concerning the facial structure of isotone projection cones as well their application to the Complementarity Problem.
This note concerns two necessary conditions for order to a cone KK in the Hilbert space HH to be an isotone projection. The main results can be summarized in the form:
THEOREM 1. If KK is a generating isotone projection cone in HH then it is latticial and correct.
It turns out (see Theorem 9 in Section 5) that our theorem above generalizes the necessary part of the main result in [4] (which is stated as Theorem 9 in Section 5).
3. Isotone Projection Cones Are Latticial
The cone KK in HH is called subdual if K subK^(**)K \subset K^{*}. We say that KK is normal if there exists a constant gamma > 0\gamma>0 such that 0 <= x <= y0 \leqslant x \leqslant y implies gamma||x|| <= ||y||\gamma\|x\| \leqslant\|y\|.
Obviously if HH is ordered by KK and for every u,v in Hu, v \in H satisfying 0 <= u <= v0 \leqslant u \leqslant v we have that ||u|| <= ||v||\|u\| \leqslant\|v\| we deduce that KK is normal.
The ordered space HH and its positive cone KK is called regular if every decreasing sequence of elements in KK is convergent.
This implies in particular that each increasing order bounded sequence in HH is convergent.
Proposition 2. If KK is an isotone projection cone in the Hilbert space HH then it is normal and regular.
Proof. In our paper [5] we proved (Theorem 2.2) that every isotone projection cone is subdual.
We prove now that KK is normal. Indeed, suppose 0 <= u <= v0 \leqslant u \leqslant v. Then v-u in Kv-u \in K and since KK is subdual, v-u inK^(**)v-u \in K^{*} and we have (:v-u,v:) >= 0\langle v-u, v\rangle \geqslant 0 and (:v-u,u:) >= 0\langle v-u, u\rangle \geqslant 0 whence it follows that ||u||^(2) <= (:u,v:) <= ||v||^(2)\|u\|^{2} \leqslant\langle u, v\rangle \leqslant\|v\|^{2}, which shows that KK is normal.
A result of McArthur [7] shows that every closed normal cone in a Banach space is regular if this space does not contain any subspace isomorphic with c_(0)c_{0} (the Banach space of all sequences of real numbers converging to zero equipped with the supremum norm). Since this holds for a Hilbert space we conclude that KK is also regular.
Proposition 3. Every generating isotone projection cone KK in the Hilbert space HH is latticial.
Proof. We begin by proving the following auxiliary result.
Lemma 4. Let KK be a closed generating cone in HH.
If there exist a in u+K,b in v+Ka \in u+K, b \in v+K with the properties a=P_(u+K)(b)a=P_{u+K}(b), b=P_(v+K)(a)b=P_{v+K}(a), then a=b in(u+K)nn(v+K)a=b \in(u+K) \cap(v+K).
Proof of the Lemma. Since KK is generating there exists an element w in(u+K)nn(v+K)w \in(u+K) \cap(v+K). (This is equivalent with the assertion that there exists an element ww with u <= wu \leqslant w and v <= wv \leqslant w.
Since KK is generating, there exist u_(1),u_(2),v_(1),v_(2)in Ku_{1}, u_{2}, v_{1}, v_{2} \in K such that u=u_(1)-u_(2)u=u_{1}-u_{2} and v=v_(1)-v_(2)v=v_{1}-v_{2}.
Hence u <= u_(1),v <= v_(1)u \leqslant u_{1}, v \leqslant v_{1} and as u_(1)u_{1} and v_(1)v_{1} are in KK it follows that u_(1) <= u_(1)+v_(1)u_{1} \leqslant u_{1}+v_{1} and v_(1) <= u_(1)+v_(1)v_{1} \leqslant u_{1}+v_{1}. Thus we can take w=u_(1)+v_(1)w=u_{1}+v_{1} to have the required property.)
We have by the characterization (1) of the projections the relations
Using the conditions in the assertion of the lemma on aa and bb, the second relation can be written in the form (:P_(v+kappa)(a)-a,a-w:) >= 0\left\langle P_{v+\kappa}(a)-a, a-w\right\rangle \geqslant 0. We have on the other hand,
We now prove Proposition 3.
Let uu and vv be arbitrary elements in HH. We shall show using the isotone projection property of KK that they admit a least upper bound u vv vu \vee v by constructing effectively this element. If uu and vv are comparable we have nothing to do.
Suppose they are not comparable.
Let ww be an arbitrary upper bound of the set {u,v}\{u, v\}, i.e., an arbitrary element of (u+K)nn(v+K)(u+K) \cap(v+K) which is not empty since KK is generating by hypothesis.
We see next that if P_(K)P_{K} is isotone, then P_(y+K)P_{y+K} is isotone too for an arbitrary yy in HH. This follows directly from the relation P_(y+K)(x)=P_(K)(x-y)+yP_{y+K}(x)= P_{K}(x-y)+y, which can be verified directly using (1). Hence P_(u+K)P_{u+K} and P_(v+K)P_{v+K} are both isotone.
Since neither of the convex sets u+Ku+K and v+Kv+K is contained in the other, using Lemma 4 we conclude that the relations u=P_(u+K)(v)u=P_{u+K}(v) and v=P_(v+K)(u)v=P_{v+K}(u) cannot hold simultaneously.
Suppose u!=P_(u+K)(v)in u+Ku \neq P_{u+K}(v) \in u+K.
Hence u <= P_(u+K)(v) <= P_(u+K)(w)=wu \leqslant P_{u+K}(v) \leqslant P_{u+K}(w)=w, since P_(u+K)P_{u+K} is isotone.
Let us consider the operators Q:=P_(v+K)@P_(u+K)Q:=P_{v+K} \circ P_{u+K} and R:=P_(u+K)@P_(v+K)R:=P_{u+K} \circ P_{v+K}. They are isotone since P_(u+K)P_{u+K} and P_(v+K)P_{v+K} are. Put v_(n)=Q^(n)(v),u_(1)=P_(u+K)(v)v_{n}=Q^{n}(v), u_{1}=P_{u+K}(v), and u_(n)=R^(n-1)(u_(1))u_{n}=R^{n-1}\left(u_{1}\right). Then we have the relations
since u <= u_(1),v <= v_(1),P_(u+K),Qu \leqslant u_{1}, v \leqslant v_{1}, P_{u+K}, Q, and RR are isotone and since P_(u+K)(w)=Q(w)=R(w)=wP_{u+K}(w)= Q(w)=R(w)=w.
(Obviously P_(v+K)@P_(u+K)(v)in v+K,quadP_{v+K} \circ P_{u+K}(v) \in v+K, \quad hence quad v <= P_(v+K)@P_(u+K)(v)=Q(v)=v_(1)quad\quad v \leqslant P_{v+K} \circ P_{u+K}(v)= Q(v)=v_{1} \quad and quadu_(1)=P_(u+K)(v) <= P_(u+K)@P_(v+K)@P_(u+K)(v)\quad u_{1}=P_{u+K}(v) \leqslant P_{u+K} \circ P_{v+K} \circ P_{u+K}(v), that is, u_(1) <= R(u_(1))=u_(2)u_{1} \leqslant R\left(u_{1}\right)=u_{2}, etc.)
Since the sequences {u_(n)}\left\{u_{n}\right\} and {v_(n)}\left\{v_{n}\right\} are increasing and majorized by ww, and since KK is regular by Proposition 2 there exist the limits
{:(8)u <= u_(0) <= w quad" and "quad v <= v_(0) <= w",":}\begin{equation*}
u \leqslant u_{0} \leqslant w \quad \text { and } \quad v \leqslant v_{0} \leqslant w, \tag{8}
\end{equation*}
since u <= u_(n) <= wu \leqslant u_{n} \leqslant w and v <= v_(n) <= wv \leqslant v_{n} \leqslant w for every nn and since KK is closed.
From the continuity of projections the relations (5), (6), and (7) imply v_(0)=P_(v+K)(u_(0))v_{0}=P_{v+K}\left(u_{0}\right) and u_(0)=P_(u+K)(v_(0))u_{0}=P_{u+K}\left(v_{0}\right).
Using Lemma 4 again we deduce that u_(0)=v_(0)in(u+K)nn(v+K)u_{0}=v_{0} \in(u+K) \cap(v+K). Since the upper bound ww is arbitrary, from the relations (8) we obtain that in fact u_(0)=v_(0)=u vv vu_{0}=v_{0}=u \vee v and the proposition is proved.
We remark that the sequences {u_(n)}\left\{u_{n}\right\} and {v_(n)}\left\{v_{n}\right\} can be stationary from a certain point onwards. (This can occur only when P_(u+K)(v)in v+KP_{u+K}(v) \in v+K.) But this case is also included in the above schema of the proof.
4. Isotone Projection Cones Are Correct
Proposition 5. For every face FF of the isotone projection cone KK in HH the space bar(sp)F\overline{s p} F projects by P_(K)P_{K} onto bar(F)\bar{F} and bar(F)\bar{F} is an isotone projection cone in bar(sp)F\overline{s p} F.
Proof. Suppose that FF is a face of KK and consider z in bar(sp)Fz \in \overline{s p} F. Then z=x-yz=x-y with x,y in F sub Kx, y \in F \subset K, whence z <= xz \leqslant x.
Since P_(K)P_{K} is isotone, one has 0 <= P_(K)(z) <= P_(K)(x)=x in F0 \leqslant P_{K}(z) \leqslant P_{K}(x)=x \in F, hence P_(K)(z)in FP_{K}(z) \in F.
Consider now z in widehat(sp)Fz \in \widehat{s p} F. Then there exists a sequence {z_(n)}\left\{z_{n}\right\} with z_(n)in widehat(sp)Fz_{n} \in \widehat{s p} F such that lim_(n rarr oo)z_(n)=z\lim _{n \rightarrow \infty} z_{n}=z.
We have shown in the above paragraph that P_(K)(z_(n))in FP_{K}\left(z_{n}\right) \in F for each nn, hence P_(K)(z_(n))rarrP_(K)(z)in bar(F)P_{K}\left(z_{n}\right) \rightarrow P_{K}(z) \in \bar{F} by continuity of P_(K)P_{K} [10]. This concludes at once that bar(sp)F\overline{s p} F projects onto bar(F)\bar{F} by P_(K)P_{K}, that P_(K∣ bar(sp)F)=P_( bar(F)∣ bar(sp)F)P_{K \mid \overline{s p} F}=P_{\bar{F} \mid \overline{s p} F}, and that P_( bar(F))P_{\bar{F}} is an isotone projection with respect to the order induced by bar(F)\bar{F} in bar(sp)F\overline{s p} F.
Remark. This proposition constitutes a generalization of Proposition 5.1 in [5].
Proposition 6. Every isotone projection cone in HH is correct.
Proof. Assume the contrary. Then there exists a face FF of KK and an clement kk of KK such that z:=P_( bar(sp)F)(k)!in Kz:=P_{\overline{s p} F}(k) \notin K.
Denote z_(0):=P_(K)(z)z_{0}:=P_{K}(z). Since z in bar(Sp)Fz \in \overline{S p} F it follows by Proposition 5 that z_(0)in bar(F)z_{0} \in \bar{F}.
We shall show first that a real number t in(0,1)t \in(0,1) can be determined such that for w:=tk+(1-t)z_(0)w:=t k+(1-t) z_{0} one has
since (:z-k,z-z_(0):)=0(z-z_(0)in bar(sp)F:}\left\langle z-k, z-z_{0}\right\rangle=0\left(z-z_{0} \in \overline{s p} F\right. and z-k=P_( bar(sp)F)(k)-kz-k=P_{\overline{s p} F}(k)-k is orthogonal to bar(sp)F\overline{s p} F ).
Since ||z-k|| < ||z_(0)-k||\|z-k\|<\left\|z_{0}-k\right\| by the definition of zz, it follows that putting 1-t=||z-k||^(2)//||z_(0)-k||^(2) < 11-t=\|z-k\|^{2} /\left\|z_{0}-k\right\|^{2}<1, relation (9) holds. Obviously, w in Kw \in K.
From the definition of z_(0)z_{0} we have using relation (1) that
This relation contradicts (10) and shows that our hypothesis that KK is not correct is false.
So Theorem 1 is completely proved.
5. The Finite Dimensional Case
One dimensional faces of the cone KK in Euclidean space R^(n)R^{n} are called extreme rays.
The closed and generating cone KK in R^(n)R^{n} was called thin in [4] if (:u,v:) <= 0\langle u, v\rangle \leqslant 0 for any two nonzero vectors uu and vv on any two different extreme rays of K^(0)K^{0}.
The basic result in [4] is the following.
Theorem 7. The generating cone KK in R^(n)R^{n} is an isotone projection if and only if it is thin.
How does this theorem relate to our Theorem 1?
To answer this question we prove first the following result.
Proposition 8. The closed generating cone KK in the Euclidean space R^(n)R^{n} is thin if and only if it is latticial and correct.
Proof. The latticiality of a thin cone follows from (i) in Lemma 2 of [4] and the characterization of Youdine of latticial cones in R^(n)R^{n} [11].
From (iii) of the same lemma we have that the thin cone KK and its polar K^(0)K^{0} can be represented in the form K=cone{e_(1),e_(2),dots,e_(n)},K^(0)=K=\operatorname{cone}\left\{e_{1}, e_{2}, \ldots, e_{n}\right\}, K^{0}= cone {u_(1),u_(2),dots,u_(n)}\left\{u_{1}, u_{2}, \ldots, u_{n}\right\} where conc MM is the conic hull of M,e_(i),u_(j),i,j=1,dots,nM, e_{i}, u_{j}, i, j= 1, \ldots, n, are unit vectors, and (:e_(i),u_(j):)=0,(:u_(i),u_(j):) <= 0\left\langle e_{i}, u_{j}\right\rangle=0,\left\langle u_{i}, u_{j}\right\rangle \leqslant 0 if i!=j,i,j=1,2,dots,ni \neq j, i, j= 1,2, \ldots, n.
Let PP denote the projection onto Pi(u_(n)):={x inR^(n)∣(:x,u_(n):)=0}=sp{e_(1),e_(2),dots,e_(n-1)}\Pi\left(u_{n}\right):=\left\{x \in R^{n} \mid\left\langle x, u_{n}\right\rangle=0\right\}= s p\left\{e_{1}, e_{2}, \ldots, e_{n-1}\right\}.
We show that P(K)sub KP(K) \subset K. To this end we have to show only that P(e_(n))in KP\left(e_{n}\right) \in K, since an arbitrary element xx of KK is of the form x=c_(1)e_(1)+cdots+c_(n-1)e_(n-1)+c_(n)e_(n)quadx=c_{1} e_{1}+\cdots+ c_{n-1} e_{n-1}+c_{n} e_{n} \quad with quadc_(i) >= 0,quad i=1,2,dots,n,quad\quad c_{i} \geqslant 0, \quad i=1,2, \ldots, n, \quad and quad P(x)=c_(1)e_(1)+cdots+c_(n-1)e_(n-1)+c_(n)P(e_(n))\quad P(x)=c_{1} e_{1}+\cdots+ c_{n-1} e_{n-1}+c_{n} P\left(e_{n}\right).
(since (:e_(n),u_(n):) < 0\left\langle e_{n}, u_{n}\right\rangle<0 and (:u_(n),u_(j):) <= 0\left\langle u_{n}, u_{j}\right\rangle \leqslant 0 by hypothesis), while (:P(e_(n)),u_(n):)=0\left\langle P\left(e_{n}\right), u_{n}\right\rangle=0. This is enough to see that P(e_(n))in(K^(0))^(0)=KP\left(e_{n}\right) \in\left(K^{0}\right)^{0}=K. Apply induction to sec that KK is correct.
Suppose now that KK is latticial and correct and show that it is thin.
It is immediate that KK is closed and generating. Since KK is latticial, K^(0)K^{0} is latticial too [9], and hence it is a cone hull of nn unit vectors u_(1),u_(2),dots,u_(n)u_{1}, u_{2}, \ldots, u_{n} which generate all the extreme rays of K^(0)K^{0} (this follows from the theorem of Youdine cited above).
Assume that KK is not thin. Then there exists at least a pair of vectors in the set {u_(1),u_(2),dots,u_(n)}\left\{u_{1}, u_{2}, \ldots, u_{n}\right\} which form an acute angle. Suppose for instance that (:u_(1),u_(2):) > 0\left\langle u_{1}, u_{2}\right\rangle>0.
The faces F_(i):={x in K∣(:x,u_(i):)=0},i=1,2F_{i}:=\left\{x \in K \mid\left\langle x, u_{i}\right\rangle=0\right\}, i=1,2, are different (n-1)(n-1)-dimensional faces of KK.
Their spans are Pi(u_(i)):={x in H∣(:x,u_(i):)=0},i=1,2\Pi\left(u_{i}\right):=\left\{x \in H \mid\left\langle x, u_{i}\right\rangle=0\right\}, i=1,2. Pick x inF_(1)\\F_(2)x \in F_{1} \backslash F_{2} and project it into Pi(u_(2))\Pi\left(u_{2}\right). Since u_(1)u_{1} is of unit norm by hypothesis, the projection ww of xx into Pi(u_(1))\Pi\left(u_{1}\right) is of form w=x-(:x,u_(1):)u_(1)w=x-\left\langle x, u_{1}\right\rangle u_{1}.
and from the definition of xx, we must have (:x,u_(2):)=0,(:x,u_(1):) < 0\left\langle x, u_{2}\right\rangle=0,\left\langle x, u_{1}\right\rangle<0.
Hence we have (:w,u_(2):)=(:x,u_(2):)-(:x,u_(1):)(:u_(1),u_(2):)=-(:x,u_(1):)(:u_(1),u_(2):) > 0\left\langle w, u_{2}\right\rangle=\left\langle x, u_{2}\right\rangle-\left\langle x, u_{1}\right\rangle\left\langle u_{1}, u_{2}\right\rangle=-\left\langle x, u_{1}\right\rangle \left\langle u_{1}, u_{2}\right\rangle>0.
This relation shows by (11) that w!in Kw \notin K, that is, P_( bar(sp)F_(1))(K)⊄KP_{\overline{s p} F_{1}}(K) \not \subset K and hence KK cannot be correct.
Proposition 8 allows us to give the following equivalent formulation of Theorem 7.
Theorem 9. The generating cone KK in R^(n)R^{n} is an isotone projection if and only if it is latticial and correct.
Remarks. (1) We have in the proof of Proposition 3 a very interesting constructive method of defining the element s u p(u,v)(u vv v)\sup (u, v)(u \vee v) where u,v in Hu, v \in H and HH is ordered by an isotone projection cone.
Given an ordered Hilbert space (H,(::),, <= )(H,\langle\rangle,, \leqslant) where the order " <=\leqslant " is defined by an isotone projection cone, then for every u,v in Hu, v \in H (if uu and vv are not comparable) then the element u vv vu \vee v is the limit of a sequence obtained by an iterative method. (See the proof of Proposition 3.)
(2) We recall [6,2] that a Hilbert lattice is a Hilbert space ( H,(:H,\langle,:))orderedbyaclosedpointedconvexconeK\rangle ) ordered by a closed pointed convex cone K such that: (1) HH is a vector lattice; (2) ||x|| <= ||y||\|x\| \leqslant\|y\|, whenever 0 <= x <= y0 \leqslant x \leqslant y; and (3) for every x in Hx \in H, |||x|||=||x||\||x|\|=\|x\| (that is, the norm is absolute).
We note that an interesting study of absolute norms can be found in [2].
In our paper [5] we proved that if K sub HK \subset H is a self-dual cone (that is, {:K=K^(**))\left.K=K^{*}\right), then KK is an isotone projection if and only if ( H,(:H,\langle,:))isa\rangle ) is a Hilbert lattice with respect to the order defined by KK.
From Theorem 1 and Proposition 2 we deduce that if (H,(::)(H,\langle\rangle,)isa) is a Hilbert space ordered by an isotone projection cone KK such that K!=K^(**)K \neq K^{*} then the norm ||||\|\| of HH is not absolute.
We finish this note with the following problem:
In an infinite dimensional Hilbert space, is a generating, latticial, and correct cone also an isotone projection cone?
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